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## Overview and History

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### CSC 421: Algorithm Design & Analysis Spring 2013 Greedy algorithms greedy algorithms examples: optimal change, job scheduling Prim's algorithm (minimal spanning tree) – PowerPoint PPT presentation

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Title: Overview and History

1
CSC 421 Algorithm Design Analysis Spring 2013
• Greedy algorithms
• greedy algorithms
• examples optimal change, job scheduling
• Prim's algorithm (minimal spanning tree)
• Dijkstra's algorithm (shortest path)
• Huffman codes (data compression)
• applicability

2
Greedy algorithms
• the greedy approach to problem solving involves
making a sequence of choices/actions, each of
which simply looks best at the moment
• local view choose the locally optimal option
• hopefully, a sequence of locally optimal
solutions leads to a globally optimal solution
• example optimal change
• given a monetary amount, make change using the
fewest coins possible
• amount 16 coins?
• amount 96 coins?

3
Example greedy change
• while the amount remaining is not 0
• select the largest coin that is ? the amount
remaining
• add a coin of that type to the change
• subtract the value of that coin from the amount
remaining
• e.g., 96 50 25 10 10 1
• will this greedy algorithm always yield the
optimal solution?
• for U.S. currency, the answer is YES
• for arbitrary coin sets, the answer is NO
• suppose the U.S. Treasury added a 12 coin
• GREEDY 16 12 1 1 1 1 (5 coins)
• OPTIMAL 16 10 5 1 (3 coins)

4
Example job scheduling
• suppose you have a collection of jobs to execute
and know their lengths
• want to schedule the jobs so as to minimize
waiting time
• Job 1 5 minutes Schedule 1-2-3 0 5 15
20 minutes waiting
• Job 2 10 minutes Schedule 3-2-1 0 4 14 18
minutes waiting
• Job 3 4 minutes Schedule 3-1-2 0 4 9
13 minutes waiting
• GREEDY ALGORITHM do the shortest job first
• i.e., while there are still jobs to execute,
schedule the shortest remaining job

does the greedy approach guarantee the optimal
schedule? efficiency?
5
Application minimal spanning tree
• consider the problem of finding a minimal
spanning tree of a graph
• a spanning tree of a graph G is a tree (no
cycles) made up of all the vertices and a subset
of the edges of G
• a minimal spanning tree for a weighted graph G is
a spanning tree with minimal total weight
• minimal spanning trees arise in many real-world
applications
• e.g., wiring a network of computers connecting

spanning tree? minimal spanning tree?
example from http//compprog.wordpress.com/
6
Prim's algorithm
• to find a minimal spanning tree (MST)
• select any vertex as the root of the tree
• repeatedly, until all vertices have been added
• find the lowest weight edge with exactly one
vertex in the tree
• select that edge and vertex and add to the tree

7
Prim's algorithm
• to find a minimal spanning tree (MST)
• select any vertex as the root of the tree
• repeatedly, until all vertices have been added
• find the lowest weight edge with exactly one
vertex in the tree
• select that edge and vertex and add to the tree

minimal spanning tree? is it unique?
8
Correctness of Prim's algorithm
• Proof (by induction) Each subtree T1, T1, ,
TV in Prim's algorithm is contained in a MST.
Thus, TV is a MST.
• BASE CASE T1 contains a single vertex, so is
contained in a MST.
• ASSUME T1, , Ti-1 are contained in a MST.
• STEP Must show Ti is contained in a MST.
• Assume the opposite, that Ti is not contained in
a MST.
• Let ei be the new edge (i.e., minimum weight edge
with exactly one vertex in Ti-1).
• Since we assumed Ti is not part of any MST,
adding ei to a MST will yield a cycle.
• That cycle must contain another edge with exactly
one vertex in Ti-1 .
• Replacing that edge with ei yields a spanning
tree, and since ei had the minimal weight of any
edge with exactly one vertex in Ti-1, it is a
MST.
• Thus, Ti is contained in a MST ? CONTRADICTION!

9
Efficiency of Prim's algorithm
• brute force (i.e., adjacency matrix)
• simple (conservative) analysis
• for each vertex, must select the least weight
edge ? O(V E)
• more careful analysis
• note that the number of eligible edges is
shrinking as the tree grows
• S (V deg(vi)) O(V2 E) O(V2)
• smarter implementation
• use a priority queue (min-heap) to store
vertices, along with minimal weight edge
• to select each vertex remove from PQ ? V
O(log V) O(V log V)
• to update each adjacent vertex after removal (at
most once per edge)
• E O(log V) O(E log V)
• overall efficiency is O( (EV) log V )

10
Application shortest path
• consider the general problem of finding the
shortest path between two nodes in a graph
• flight planning and word ladder are examples of
this problem
• - in these cases, edges have uniform cost
(shortest path fewest edges)
• if we allow non-uniform edges, want to find
lowest cost/shortest distance path

Redville ? Purpleville ?
example from http//www.algolist.com/Dijkstra's_al
gorithm
11
Modified BFS solution
• we could modify the BFS approach to take cost
into account
end (i.e., queue), add in order of path cost
(i.e., priority queue)
• Redville0
• Redville, Blueville5,
• Redville, Orangeville8,
• Redville, Greenville10
• Redville, Orangeville8,
• Redville, Blueville, Greenville8,
• Redville, Greenville10,
• Redville, Blueville, Purpleville12
• Redville, Blueville, Greenville8,
• Redville, Greenville10,
• Redville, Orangeville, Purpleville10,
• Redville, Blueville, Purpleville12

note as before, requires lots of memory to store
all the paths HOW MANY?
12
Dijkstra's algorithm
• alternatively, there is a straightforward greedy
algorithm for shortest path
• Dijkstra's algorithm
• Begin with the start node. Set its value to 0 and
the value of all other nodes to infinity. Mark
all nodes as unvisited.
• For each unvisited node that is adjacent to the
current node
• If (value of current node value of edge) lt
(value of adjacent node), change the value of the
• Otherwise leave the value as is.
• Set the current node to visited.
• If unvisited nodes remain, select the one with
smallest value and go to step 2.
• If there are no unvisited nodes, then DONE.
• this algorithm is O(N2), requires only O(N)

13
Dijkstra's algorithm example
• suppose want to find shortest path from Redville
to Purpleville
1. Begin with the start node. Set its value to 0 and
the value of all other nodes to infinity. Mark
all nodes as unvisited
• For each unvisited node that is adjacent to the
current node
• If (value of current node value of edge) lt
(value of adjacent node), change the value of the
• Otherwise leave the value as is.
• Set the current node to visited.

14
Dijkstra's algorithm example cont.
• If unvisited nodes remain, select the one with
smallest value and go to step 2.
• Blueville set Greenville to 8 and Purpleville to
12 mark as visited.
• Greenville no unvisited neighbors mark as
visited.
• If unvisited nodes remain, select the one with
smallest value and go to step 2.
• Orangeville set Purpleville to 10 mark as
visited.
• If there are no unvisited nodes, then DONE.

With all nodes labeled, can easily construct the
shortest path HOW?
15
Correctness efficiency of Dijkstra's algorithm
• analysis of Dijkstra's algorithm is similar to
Prim's algorithm
• can show that each greedy selection is safe,
• brute force (i.e., adjacency matrix) approach
• for each vertex, need to select shortest edge ?
O(V E)
• or, more carefully, S (V deg(vi)) O(V2
E) O(V2)
• smarter implementation
• use a priority queue (min-heap) to store
vertices, along with minimal weight edge
• to select each vertex remove from PQ ? V
O(log V) O(V log V)
• to update each adjacent vertex after removal ?
E O(log V) O(E log V)
• overall efficiency is O( (EV) log V )

16
Another application data compression
• in a multimedia world, document sizes continue to
increase
• a 6 megapixel digital picture is 2-4 MB
• an MP3 song is 3-6 MB
• a full-length MPEG movie is 800 MB
• storing multimedia files can take up a lot of
disk space
requires significant bandwidth
• it could be a lot worse!
• image/sound/video formats rely heavily on data
compression to limit file size e.g., if no
compression, 6 megapixels 3 bytes/pixel 18
MB
• the JPEG format provides 101 to 201 compression
without visible loss

17
Audio, video, text compression
• audio video compression algorithms rely on
domain-specific tricks
• lossless image formats (GIF, PNG) recognize
repeating patterns (e.g. a sequence of white
pixels) and store as a group
• lossy image formats (JPG, XPM) round pixel values
and combine close values
• video formats (MPEG, AVI) take advantage of the
fact that little changes from one frame to next,
so store initial frame and changes in subsequent
frames
• audio formats (MP3, WAV) remove sound out of
hearing range, overlapping noises
• in the absence of domain-specific knowledge,
can't do better than a fixed-width code
• e.g., ASCII code uses 8-bits for each character
• '0' 00110000 'A' 01000001 'a' 01100001
• '1' 00110001 'B' 01000010 'b' 01100010
• '2' 00110010 'C' 01000011 'c' 01100011
• . . .
• . . .
• . . .

18
Fixed- vs. variable-width codes
• suppose we had a document that contained only the
letters a-f
• with a fixed-width code, would need 3 bits for
each character
• a 000 d 011
• b 001 e 100
• c 010 f 101
• if the document contained 100 characters, 100 3
300 bits required
• however, suppose we knew the distribution of
letters in the document
• a45, b13, c12, d16, e9, f5
• can customize a variable-width code, optimized
for that specific file
• a 0 d 111
• b 101 e 1101
• c 100 f 1100
• requires only 451 133 123 163 94
54 224 bits

19
Huffman codes
• Huffman compression is a technique for
constructing an optimal variable-length code
for text
• optimal in that it represents a specific file
using the fewest bits
• (among all symbol-for-symbol codes)
• Huffman codes are also known as prefix codes
• no individual code is a prefix of any other code
• a 0 d 111
• b 101 e 1101
• c 100 f 1100
• this makes decompression unambiguous
1010111110001001101
• note since the code is specific to a particular
file, it must be stored along with the compressed
file in order to allow for eventual decompression

20
Huffman trees
• to construct a Huffman code for a specific file,
utilize a greedy algorithm to construct a Huffman
tree
• process the file and count the frequency for each
letter in the file
• create a single-node tree for each letter,
labeled with its frequency
• repeatedly,
• pick the two trees with smallest root values
• combine these two trees into a single tree whose
root is labeled with the sum of the two subtree
frequencies
• when only one tree remains, can extract the codes
from the Huffman tree by following edges from
root to each leaf (left edge 0, right edge 1)

21
Huffman tree construction (cont.)
the code corresponding to each letter can be read
by following the edges from the root left edge
0, right edge 1 a 0 d 111 b 101 e
1101 c 100 f 1100
22
Huffman code compression
• note that at each step, need to pick the two
trees with smallest root values
• perfect application for a priority queue
(min-heap)
• store each single-node tree in a priority queue
(PQ) O(N log N)
• repeatedly, O(N) times
• remove the two min-value trees from the PQ O(log
N)
• combine into a new tree with sum at root and
insert back into PQ O(log N)
• total efficiency O(N log N)
• while designed for compressing text, it is
interesting to note that Huffman codes are used
in a variety of applications
• the last step in the JPEG algorithm, after
image-specific techniques are applied, is to
compress the resulting file using a Huffman code
• similarly, Huffman codes are used to compress
frames in MPEG (MP4)

23
Greed is good?
• IMPORTANT the greedy approach is not applicable
to all problems
• but when applicable, it is very effective (no
planning or coordination necessary)